Question

Cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρt . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a **

T = 2π √hρ/ρtg

where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Answer 1

Weight of the cork = Weight of the liquid displaced by the floating cork

Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides the restoring force to the cork.

Up-thrust = Restoring force, F = Weight of the extra water displaced

F = –(Volume × Density × g)

Volume = Area × Distance through which the cork is depressed

Volume = Ax

F = – A x ρt g … (i)

According to the force law: